I'll break down affinity approaches according to seeding, early game, mid-game and late-game:
Seeding:
If you want to pick a certain affinity before you land, some of the space gear might help. Lifeform sensor will help you find xenomass and nests and you could use it to rule harmony in or out. Tectonic scanner is very useful as a general approach if you are thinking of going purity, but might choose other affinities if the surrounding environment recommends it because of titanium's high production. If you happen to start with titanium near your capital, then you have a brilliant start. It kind of makes engineers redundant if you think about it.
Early game:
If there appears to be no clear choice for an affinity, whether it be limited affinity strategic resources near you, keep exploring and eventually you will find something that encourages you to pick a particular affinity. While some might prefer to roleplay and they end up picking supremacy when there is no viable firaxite near them, if you one of those people, then you would be relying solely on your basic units. It would also mean level 4 zerging would be off the table until you are level 6 or 7 at least.
On the other hand, if you are spoiled for choice, then it comes down to comparing numbers of strategic resources and figuring out which affinity your opponent(s) is going to pick. You could have a case where harmony becomes attractive due to open lands which you could spray with miasma, but if your opponent(s), namely your neighbour(s), is going to pick harmony, then try another affinity as miasma would also help them attack you if they want to.
Mid game:
At this point, you would be around turn 100-150 in quick games and turn 130-180 in standard picked games. You should have a few levels in your preferred affinity. If there are any affinity quests to do, complete them. If you happen to get a deep space telescope affinity quest, completing it would ultimately depend on your willingness to pursue a contact victory. If not, then getting the techs might not be worth it.
Keeping an eye on your opponent's affinity levels is crucial here. If you have less affinity levels than your neighbour (around 2+ level difference or more) then there is a good chance they will consider attacking you. You should prepare for the worst if they hit levels 4, 6, 8, 11 and 13 respectively. Level 4 brings the unique units, level 6 unlocks level 3 infantry and better planes, level 8 unlocks level 3 tanks, level 11 unlocks max upgraded infantry and better planes and level 13 unlocks hovertanks. If you are doing an island map, levels 5, 10, 13 and 15 will be your greatest concern.
Also keep an eye out on their strategic resources. If they have tons of their preferred strategic resource relative to their affinity, they could mass produce their unique units or provide at least several mega units, with the xenotitan being the most dangerous of them. So to conclude, mid-game is when the arms/affinity race begins and you should also aim for your preferred victory at this stage, but preferably as early as possible.
Late game:
Forget the awesome battlefields of Modern Warfare and Battlefield 3. Battles in this game combine the mythical clash of the titans, modern warfare and sci-fi warfare similar to star wars (on land of course) and other sci-fi series all in this strategic game form. You know you are in late game if you can build epic units and the turn number has passed 200 for both standard and quick games. At this point, it becomes important to know how your soldiers fight the most effectively.
As harmony, your role is to spray miasma using condensors as mainly a defence mechanism. Do not consider this approach if you have a neighbour following harmony as well until they are dealt with. Your soldiers are very stubborn and loyal only to their fellow soldiers, in that they have combat bonuses when they are alone. They are willing to fight to the death, and even suicide bomb themselves for the greater good. Combined with miasma and extensive mutations to the human body, your soldiers will heal rapidly and fight viciously in a rather spread out formation. Along with this, you have your xenotitan, the strongest melee unit in the game. Rushing down purity to turn it into a true xenotitan will provide with much needed ranged resistance. Some of your ranged units can also heal on kill and have the chance to interrupt enemy formations with the scatter effect, where you hit your enemy so hard they are either knocked back into another tile or they run for their life for 1 tile and then regain their senses.
As purity, your role is to focus on defence until you can get to level 13. At that point, you choose between domination or the other peaceful victories available to you (exodus and contact). Your soldiers develop incredible improvements to converting energy from movement to combat and they are most effective when they have spare moves. If you wish to go on the offensive, you will need your level 4 hovertanks which have the option of being upgraded with city attack bonuses. If you wish to turtle, which is mainly what purity is about, your wardens and aeges (plural for aegis) can have combat bonuses with garrisoned in a city. Your navy also operates strongly on movement-to-strength combat bonuses as well as bonus range. Just make sure your skies are clear from satellites as you turtle to victory or spam hovertanks for domination victory.
As supremacy, you do not call yourself a human individual, you call yourself an autonomous member of a great society. Your soldiers operate effectively when together and can become irresistible in this way. You also have nodes giving you impressive healing bonuses when defending. Your planes and rovers can also gain a strong combat bonus against wounded units, making them very efficient in finishing off enemies. The rovers, moreover, can get stronger flanking bonuses (NB: don't get ignore ZoCs, they already have that). Your frigates can also hold planes or can hit and run like sea-keshiks. Your obrital range becomes drastically expanded at level 8 and once you get phasal transporters, you can teleport units from halfway across the world.
With either affinity, don't forget to use satellites when you can. If you are low or have no titanium or oil resources, consider getting tacnet nubs to defend or attack.
I hope this guide helps.